1. Introduction

As candidates for subwavelength confinement, both plasmon [1–5] and all-dielectric [6–10] periodic nanoparticle chains have been proposed. However, both of these approaches are frequency-sensitive, disorder-sensitive or incident-angle-sensitive [11–14]. These approaches cannot meet the broadband requirement for photonic integrated circuits [15]. In this work, we demonstrate that with the use of all-angle self-collimation, an ultrathin chain of periodic subwavelength nanoparticles can be designed to route and manipulate light at the subwavelength scale and is broadband, incident-angle tolerant, and robust against disorder (including size disorder and position disorder). In addition, the ultrathin chain can be used to sharply bend and split light beams below the diffraction limit. Moreover, our nanostructure is simple and compact and can be compatible with standard CMOS technology. These advantages demonstrate that the self-collimation has great potential for use in silicon-based photonic circuits.

2. Results

All-angle self-collimation of periodic nanostructures. To describe the ability of periodic nanostructures to overcome the diffraction limit in a relatively general form, we first construct a physical picture for the all-angle self-collimation process.

A road map from the self-collimation of PCs to all-angle self-collimation is shown in Figs. 1(a)-1(d). Let us consider a two-dimensional (2D) rod-type PC with a rectangular lattice, as shown in the inset of Fig. 1. We find that the lattice parameter β = b/a plays a key role in the collimation effect. From Figs. 1(a)-1(d), we can see that as β increases, the equi-frequency contours (EFCs) become increasingly flat in the high k_x region. These flat EFCs correspond to self-collimation. Moreover, flat EFCs across the entire zone with a broad bandwidth can be obtained. In other words, wide angle and broadband self-collimation can be achieved with a larger β. The field distributions are shown in Figs. 1(e)-1(h). From these figures, we can see the beam becomes non-diffraction. When β becomes increasingly large, the above-mentioned effect becomes increasingly obvious [15]. As a result, the light beams are collimated and propagate along one array of the PCs without diffraction, i.e., nearly all of the electromagnetic fields for an arbitrary incident angle are compressed and routed at the array, which appears as a bright chain, as shown in Fig. 2(a). In this case, according to the photonic bandgap engineering theory, we can remove the other arrays in the PCs where the fields are much weaker, leaving only a single array of nanorods to route light, as shown in Fig. 2(b). Obviously, this operation does not have much impact on the propagation of light, which can also be seen by the comparison between Fig. 2(a) and Fig. 2(b). Hence, the single chain of nanorods can also be used to compress and route light, for example, the self-collimation PCs with a large β. Consequently, the single array that has a special self-collimation effect can be regarded as the special PC with β~∞, and we refer to such a special self-collimation together with traditional self-collimation as all-angle self-collimation. Experimental observation verifies the phenomenon of this self-collimation in a single chain of silicon nanorods, as shown in Fig. 2(c) and 2(d).

 

Fig. 1 (a-d) The EFCs of photonic crystals with β = 1, 2, 3 and 6, respectively. The EFCs are calculated with plane wave expansion method. The insets are the corresponding PCs. (e-h) The E field intensity distribution when a point light source is used to excite the photonic crystals when β = 1, 2, 3 and 6. The TM (E_y polarized) point source is placed just close to the first rod on the left. The refractive indices of the rod is 3.5. The radius of the rods is r = 160 nm and the separation of the nearest rods is a = 400nm. For (h), the propagation loss is 10.7 dB/mm. (i) The dispersion relation of a single array of nanorods. The normalized frequency 0.258 corresponds to wavelength 1.55μm. The two bands also indicate a broadband characteristic which covers a wavelength range larger than 0.7μm.

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Fig. 2 The E field distribution of a self-collimated beam in (a) a photonic crystal (with β = 3) and (b) the corresponding single nanorod chain. (c) The side-view SEM image for the fabricated single nanorod chain. (d) The ray trace of the propagation light in the nanorod chain. The trace is captured with an infrared camera set above the chain.

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Besides 2D all-dielectric periodic structures, all-angle self-collimation can also be observed in plasmonic periodic structures and in three-dimensional (3D) periodic structures.

Compressing and routing light via all-angle self-collimation. Fig. 1(d) shows that the EFCs of the single chain of particles with all-angle self-collimation (here, we use the PC that has a large β (β = 6) to approximate the EFCs of the single array of particles) have a very wide frequency region in which the EFCs are very flat, with the group velocity of light along the y direction v_gy = ∂ω/∂k_y '≈0. Therefore, light can only propagate along the x direction rather than the z direction, i.e., the collimation effect occurs. In addition, when the collimated light beam propagates along the single array of particles, the beam width can be compressed to below subwavelength size because the lattice constant is small enough. From Fig. 1(d), the collimation effect is found to be broadband in the frequency range from ω = 0.12 to 0.2 (in units of 2πc/a). In addition, the flat EFCs are found across the entire Brillouin zone, the fact is, the collimation is angle-tolerant, which allows the light beams to directly couple to the single chain at nearly all incident angles. Furthermore, we predict that the self-collimation process is disorder-tolerant because it is based on the average effect of multi-scattering processes. In summary, the self-collimation is predicted to be nearly all-angle tolerant, broadband, and robust against disorder, thereby enabling light to be compressed and routed by a single array of particles beyond the diffraction limit.

To show the angle-tolerance of the all-angle self-collimation phenomenon, we use light beams incident to the single array of particles with different large incident angles. Figures 3(a) and 3(b) show the numerical results based on FDTD simulations [16], indicating that Gaussian beams are incident at angles of 45° and 75° in the single nanoparticle chain, respectively. The normalized coupling efficiency to different incident angles in shown Fig. 3(c). The coupling efficiency is good with an incident angle smaller than 55°, remaining more than 76%. The coupling efficiency continues to decrease to 45% when the incident angle increases to 75°. From these figures, we can see the light beams are directly coupled to the chain and are then compressed and routed along the chain. In addition to the numerical simulations, experimental observations also verify our prediction, as shown in Figs. 3(d) and 3(e). The parameters of the dielectric nanorod chain are designed to be a₀ = 400 nm, r = 160 nm and the height of the nanoparticle h = 810 nm.

 

Fig. 3 FDTD simulations for propagation of light beams at wavelength λ = 1.55 µm with large incident angles: (a) 45° incidence; (b) 75° incidence. The zigzag distribution is due to the spinning-and-coupling propagation with oblique incident light for the 2nd band mode. (c) The normalized coupling efficiency to the incident angle. (d),(e) The ray trace of the light beam captured by an infrared camera.

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The all-angle self-collimation is predicted to be strongly robust against disorder. To demonstrate this robustness experimentally, we fabricated samples with intentional disorder as defects into our nanorod chain, including size-disorder and position-disorder of the particles. The experimental results are presented in Figs. 4(a)-4(d). In Fig. 4(a), there is a position-disorder particle with a deviation of 160 nm in the chain; the far field observation for light propagating along the chain is shown in Fig. 4(e). From this figure, we can see that the light wave can overcome the position-disorder in the chain. In Figs. 4(b), 4(c), and 4(d), the chains have a size-disorder particle with radii of 240 nm, 100 nm and 0, respectively. The corresponding far field observations are shown in Figs. 4(f)-4(h). These figures demonstrate the ability of this chain to overcome the size-disorder from the chain. Specifically, we calculated the transmission spectrum for position-disorder and radius-disorder strength in our experiments, as shown in Figs. (i) and (j). We can see that the chain is quite robust against position-disorder (the normalized transmission is larger than 0.9). These data also suggests that light could propagate in a bend rod chain, which is prove in our experiments. For radius-disorder strength, the transmission rate fluctuates from 0.5 to 1. The different energy loss is mainly caused by the scattering of the varying radius-disorder.

 

Fig. 4 Ray trace of the electromagnetic energy along the nanoparticle chain. (a) Ray trace along the chain when positional disorder is introduced with a deviation of 160 nm from the chain. (b), (c) and (d) Ray traces along the nanorod chain when rods are introduced with a radius of 240nm, 100 nm and 0, respectively. (e)-(h) The corresponding ray traces in the nanorod chains. (i)-(j) Transmission spectrum to position shift (left inset) and radius of the defect (right inset).

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The strong robustness against disorder for the single chain with all-angle self-collimation is very significant for the fabrication of such a chain. More important, from “robustness against disorder” and “nearly all-angle tolerance”, we can deduce that changing the position (or the size) of the particles does not have much impact on the light compressing and routing along the chain due to the self-collimation phenomenon. Therefore, we can tailor the positions of the particles to form a bent chain and, consequently, to bend the light sharply below the diffraction limit. Experimentally, we verify our deduction, as shown in Fig. 5(a).

 

Fig. 5 (a) The ray trace of the bending of light with the bent nanoparticle chain. (b) The ray trace of the splitter formed by the nanoparticle chain becomes two chains. (c) The letters “CAS” are formed by a nanorod chain that includes the transporting, splitting, bending, and combining functionalities.

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Similar to the light bender, producing a splitter based on chains with all-angle self-collimation is also feasible, as shown in Fig. 5(b). In addition, light combining the reverse process of light splitting is of course feasible with particle chains under this self-collimation. In Fig. 5(c), we present the letters “CAS”, which are formed by a nanorod chains that includes light routing, splitting, bending, combining functionalities (including some disorders); the letters can be observed clearly. Thus, the all-angle self-collimation provides a new method to realize the bender and the splitter of light below the diffraction limit, both of which are fundamental and essential in integrated photonics circuits.

3. Conclusion

In summary, we proposed a novel mechanism to route and manipulate electromagnetic with subwavelength nanoparticles. Based on the all-angle self-collimation, the particle chain is broadband and robust against disorder and has a wide-angle directly coupling ability. In addition, the all-angle self-collimation provides us an alternative approach for bending and splitting light. The phenomenon can be applied to both plasmonic and all-dielectric periodic nanostructures. The all-dielectric nanostructures can be designed to be CMOS-compatible and have non-intrinsic losses. The unique characteristics in the dielectric system enable it to be used in the integrated photonic circuits.

4. Method

Fabrication. Based on CMOS technology, the single chains are fabricated on silicon-on-insulator (SOI) substrates, where the top silicon layer and buried oxide layer were 810 nm and 3 μm thick, respectively. The top silicon layer was then coated with a 60-nm-thick thermal oxide layer to form a hardmask. The pattern was transferred to the hardmask via E-beam lithography, followed by reactive ion etching. Finally, the single chains are sculpted by an elaborate ICP-DRIE (inductively coupled plasma deep reactive ion etch) process.

Measurement. TM polarized light from a tunable laser source is introduced with a lensed fiber. To visualize the electromagnetic energy transport in the single chains, an infrared camera is used to capture the ray trace (via the scattered light) of the electromagnetic energy.

Simulation. The finite differential time domain (FDTD) method was utilized to conduct verifying numerical experiments.